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How to tell if something has a common factor

  1. Feb 4, 2010 #1
    if we have 6 and 9

    and we break them down to sets of prime factors {2,3} , and {3}

    if the intersection of the 2 sets is empty..does this mean that numbers have no common factors?

    or in more specific to my interest...

    if we have P and Q and we have their sets of prime factors, if the intersection is empty does this mean P/Q is an irreducible fraction?

    or is this only true for their sets of non-zero prime+composite factors?

  2. jcsd
  3. Feb 4, 2010 #2


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    Your main question has yes as an answer.

    I don't know what you mean by
  4. Feb 4, 2010 #3

    By definition, a set of prime factors won't contain any composites.
  5. Feb 4, 2010 #4


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    I'm going to guess the question and then answer what I thought the question was.

    Question: Given positive integers m and n, if the set M of primes dividing m and the set N of primes dividing n are disjoint (have an empty intersection), is m/n an irreducible fraction? [If this is false, is it at least the case that if the set M' of divisors of m and the set N' of divisors of n have {1} as their intersection, is m/n an irreducible fraction?]

    Answer: Yes, if there are no primes in common than m/n is irreducible.
  6. Feb 5, 2010 #5
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